the mass of the Sun: \(M\) which has dimension \([M]\to l^0 t^0 m^1\),

the distance from the center of the Sun at the point of closest contact, \([r]\to l^1 t^0 m^0\).

The physical constant we can think of in the first place is \(G\), which has dimension \([G]\to l^3 t^{-2} m^{-1}\). However, we now combine \(G\) and \(M\) and \(r\) to produce \(\theta\), which is identical to solving the equation

The system of equations has no nontrivial solutions. Where to find the other physical constant? We are dealing with light, thus one of the choices is the speed of light, \(c\). Add in \([c^e]\to l^e t^{-e} m^0\), we have the equations

In general, since \(\frac{G M}{r c^2}\) is dimensionless, the general form is

\[\theta = f( \frac{G M}{r c^2} ).\]

This result is already satisfactory.

Is there more we can conclude from here? Kurth took a step further and used limits. We expect \(\theta\to 0\) for small mass since we do not observe this effect in daily life. \(M\to 0\) leads to \(\frac{G M}{r c^2}\to 0\). We can even use the simplest form

\[\theta \propto \frac{G M}{r c^2} .\]

This is identical to the fact that the Taylor expansion of function \(f(x)\) at \(x\to 0\) has a neglectable zeroth order.

To summarize, we used the following techniques.

Dimensions.

Limits of physical problems compared with observations.

If GR is part of the knowledge pool, we notice that the radius \(R\) of the celestial body is not consider in this analysis. When we add in this, we find three dimensionless quantities,